Theorem iineq2d 3278

Description: Equality deduction for indexed intersection.

Hypotheses

Ref	Expression
iineq2d.1	|- (ph -> A.xph)
iineq2d.2	|- ((ph /\ x e. A) -> B = C)

Assertion

Ref	Expression
iineq2d	|- (ph -> |^|_x e. A B = |^|_x e. A C)

Proof of Theorem iineq2d

Step	Hyp	Ref	Expression
1		iineq2d.1	. . 3 |- (ph -> A.xph)
2		iineq2d.2	. . . 4 |- ((ph /\ x e. A) -> B = C)
3	2	ex 402	. . 3 |- (ph -> (x e. A -> B = C))
4	1, 3	r19.21ai 2174	. 2 |- (ph -> A.x e. A B = C)
5		iineq2 3274	. 2 |- (A.x e. A B = C -> |^|_x e. A B = |^|_x e. A C)
6	4, 5	syl 12	1 |- (ph -> |^|_x e. A B = |^|_x e. A C)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  |^|_ciin 3256

This theorem is referenced by:  iineq2dv 3280  pmapglbx 17251

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-iin 3258

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